Thank you for the report.
I didn't try the issue at my end but the way you have updated the root password could be the issue here, please set the root password as:
UPDATE mysql.user SET Password = PASSWORD('123456') WHERE User = 'root';
FLUSH PRIVILEGES;
Please test at your end once again after resetting root password and report us back if you are still having the issue.

Sorry my mistake with the password, I have typed a bit wrong example. In my real setup the password have been properly updated and is hashed in mysql.user table. Also I have tried with another user, dedicated only for it.
grant shutdown on multi@localhost identified by '123456'
Still the same result - error: 'Access denied for user 'multi'@'localhost' (using password: YES)

[8 Jun 2015 17:22]
Georgi Iovchev

another wrong example :) I have to stop typing and start pasting...
in my previous comment user creation should be: grant shutdown on *.* to multi@localhost identified by '123456';

[9 Jun 2015 8:47]
MySQL Verification Team

Thank you for the feedback.
Observed this with 5.6.25 build(5.6.24 not affected) on Oracle Linux Server release 7.0.
Thanks,
Umesh

I have the same problem. After adding some traces in mysqld_multi, I have found the problem : since 5.6.25, my_print_defaults no longer returns password in readable form (we have to use the "--show" option to obtain it). mysqld_multi call my_print_default to obtain the user and password to use with mysqladmin.
Also, to be able to stop instances with mysqld_multi, I have made a modification in the "defaults_for_group" function of mysqld_multi. I have changed the following line :
my $com= join ' ', 'my_print_defaults', @defaults_options, $group;
with :
my $com= join ' ', 'my_print_defaults -s', @defaults_options, $group;
Regards